Question

The frequency of vibration (v) of a string may depend upon length (I) of the string, tension (T) in the string and mass per unit length (m) of the string. Using the method of dimensions, derive the formula for v.

Answer 1

Let `v = K I^a T^b m^c ….(i)`
where K is dimensionless constant of
proportionality and a,b,c, are the dimensions of l, T
and m respectively to represent v.
The tension T stands for force whose
dimensions are `[M^1 L^1 T^(-2)]`
and `m = ("mass")/("length") =(M)/(L) = [M^1 L^(-1)]`
Writing the dimensions in (i), we get
`[M^0 L^0 T^(-1)] = L^a (M^1 L^1 T^(-2))^b (ML^(-1))^c`
`L^a M^b L^b T^(-2b) M^c L^(-c)`
`[M^0 L^0 T^(-1)] = M^(b+c) L^(a+b-c) T^(-2b)`
Applying the principle of hoomgeneity of
dimensions, we get
`b +c = 0 ...(ii)`
a +b - c = 0 ....(iii)
` -2b = -1 or b = (1)/(2)`
From (ii), `c = -b =-(1)/(2)`
from (iii), `a + (1)/(2) - (-(1)/(2)) = 0`
`a +(1)/(2) +(1)/(2) = 0 or a = -1`
Putting these values in (i), we get
`v = K I^(-1) T^(1//2) or v = (K)/(I) sqrt((T)/(m))`
Thisk is the required formula.